I have a male duck and two female ducks that live in a coop with my 13 chickens. My male duck's part was pecked at by chickens and is swollen and red. He's eating and drinking normally, and just swam this morning. I plan on separating him and giving him a nice, warm bath and washing him with some Dawn dish soap. It looks like his area is infected, so would it be alright to give him some penicillin or amoxicillin? If so, how many mg a day and for how long? If not, what can I give him?

What size of duck would dictate how much to give, but for prolapse with Calls I have used .25cc of injectable Baytril or Gentamycin. Have washed their parts with Nolvasan skin and wound cleanser, which has a residual antimicrobial effect. This is often sold where horse and/or cattle treatments are sold, I've used it for washing down my mares pre-foaling. It's not indicated for deep puncture wounds, so you'd have to kind of use your own judgement on that. I can tell you they use a similar soap solution at twice the strength to wash and prepare humans for surgery, which sometimes needs to be done in the event of a prolapse on ducks if this is in fact what you have going on or have about to happen.

We examined him some more, and it turns out that he is much worse than we thought. His area has an infection and he doesn't look healthy. My question is now, do we let him at least try to heal himself (which there's no guarantee he will) or do we put him out of his misery?

This is for phallus prostration (I'm paraphrasing):
He should be immediately isolated. Then, wash the penis with warm water and disinfect and apply a medicated ointment. Apply ointment daily until recovery, which may take several weeks or more. He shouldn't be allowed to mate for 3 months afterwards.

Now I will quote directly (in italics),

If the bird is valuable and his problem is discovered before the penis has become infected or dried out, recovery is possible.

PreventionThe best safeguard is to keep backup drakes for breeding purposes.

It does also state that often the end of the penis dries up and falls off, though the bird can still be fertile though ususally not for 6-12 months.

I would still probably try to treat, he may need an antibiotic because of infection. Because you know your birds, only you or a vet would be able to decide if he is suffering.

Okay, so we gave him a bath and he is separate from the rest of my flock. The swelling appears to have gone down a little bit, but not much. What ointment should I apply, Neosporin or a medicated ointment only for poultry and waterfowl? Thank you for the help.